<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:16"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2
 
<b>by </b><i><a class="ref" href="nat_3.html#T3">NAT_3:3</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="moebius1.html#E3:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_1_1"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:16_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pythtrip.html#D2">PYTHTRIP:def 2</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E3:16_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1"/>
not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/>
<a NAME="E5:16_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a>, <a class="txt" href="moebius1.html#E4:16_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E4:16_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="moebius1.html#E2:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:16_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E3:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/><a NAME="E5:16_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>;<br/><b>hence </b><a NAME="E6:16_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T14">EULER_1:14</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_1_2"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:16_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pythtrip.html#D2">PYTHTRIP:def 2</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E3:16_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_1"/>
not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:16_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1_2_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/>
<a NAME="E5:16_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a>, <a class="txt" href="moebius1.html#E4:16_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E4:16_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="moebius1.html#E2:16_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:16_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E3:16_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/><a NAME="E5:16_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>;<br/><b>hence </b><a NAME="E6:16_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T14">EULER_1:14</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
